It is proposed to investigate reduced rank decompositions of vectors of observed random variables which are in many respects similar to factor analysis, but do not involve indeterminante latent random variables. Sometimes such decompositions are, loosely, called "component analysis," but this term, as it is presently used, is too vague to permit meaningful study and assessment of the methods covered by it. A more limited and precise definition is offered which, nevertheless, covers a large class of potentially useful alternatives to factor methodology. The class of decompositions to be investigated, primarily at the theoretical, but eventually also at an empirical level, are of the form eta equals A xi plus epsilon, xi equals eta where eta is a p-variate vector of "observed" random variables yi, B a pxm (m less than p) matrix of "defining weights," xi an m-variate vector of "components" and xi a p-variate vectors of "residuals." The matrix A is a pxm matrix of regression weights for predicting the observed yi from the components xj in a multiple regression sense. Our interest in this particular class of decompositions is motivated by the factor model, which has a similar mathematical structure, but does not define xi and epsilon as linear functions of the observed vector n, so that both are to some extent indeterminate for given n, A and U. It is proposed to study such component decompositions coherently and in some depth at the theoretical level, and also to investigate certain special cases in their relation to the factor model, prior to studying the effects of substituting one method for the other empirically. It is expected that such a substitution, where appropriate, could lead to a considerable saving in computer time.